Unable to join public rooms

*This issue was originally created by [**@blind-oracle**](https://github.com/blind-oracle) at .*

### Background information

- **Dendrite version or git SHA**: #3244
- **SQLite3 or Postgres?**: PG
- **Running in Docker?**: No
- **`go version`**: 1.21.3
- **Client used (if applicable)**: Element

dendrite is running behind nginx that does TLS termination and reverse proxy.

```
location /.well-known/matrix/server {
return 200 '{ "m.server": "novg.net:443" }';
}

location /.well-known/matrix/client {
return 200 '{ "m.homeserver": { "base_url": "https://novg.net" } }';
}

location /_matrix {
proxy_pass http://127.0.0.1:8088;
proxy_set_header Host $host;
proxy_set_header X-Real-IP $remote_addr;
proxy_read_timeout 600;
}
```

### Description

- **What** is the problem: Unable to join any public rooms outside my host
- **Who** is affected:
- **How** is this bug manifesting:
- **When** did this first appear:

### Steps to reproduce

- Setup dendrite
- Register account
- Try to join some matrix.org or libera.chat rooms
- Observe infinite `Joining...` spinner

The errors 524/520 look like a Cloudflare timeouts or similar, but doing `curl` works fine to the given URLs.

In the logs:
```
Oct 28 18:19:27 novg dendrite[3205549]: time="2023-10-28T16:19:27.855043139Z" level=warning msg="Failed to join room through server" error="r.federation.SendJoin: contents=[] msg=Failed to PUT JSON (hostname \"matrix.org\" path \"/_matrix/federation/v2/send_join/!OGEhHVWSdvArJzumhm:matrix.org/$HohCRwjjZWmrs54SNMven2HXOr-0u2_WdKF0zl4zfJQ\"): code=524 wrapped=" room_id="!OGEhHVWSdvArJzumhm:matrix.org" server_name=matrix.org
Oct 28 18:19:28 novg dendrite[3205549]: time="2023-10-28T16:19:27.855146010Z" level=error msg="failed to join user \"@blind_oracle:novg.net\" to room \"!OGEhHVWSdvArJzumhm:matrix.org\" through 2 server(s): last error r.federation.SendJoin: contents=[] msg=Failed to PUT JSON (hostname \"matrix.org\" path \"/_matrix/federation/v2/send_join/!OGEhHVWSdvArJzumhm:matrix.org/$HohCRwjjZWmrs54SNMven2HXOr-0u2_WdKF0zl4zfJQ\"): code=524 wrapped="
Oct 28 18:19:28 novg dendrite[3205549]: time="2023-10-28T16:19:27.855207419Z" level=error msg="Failed to join room" error="contents=[] msg= code=524 wrapped=" room_id="#matrix:matrix.org" servers="[libera.chat]" user_id="@blind_oracle:novg.net"
```

Start with the federation SendJoin path shown in the logs and compare it with the nginx reverse-proxy configuration and timeout responses. Reproduce a join to the listed public rooms, trace the 520/524 failure, and consider the issue done when external room joins complete without the infinite spinner or federation timeout.

Written by the indexing model from the issue text.